چکیده انگلیسی

The problem of allocation of orders for custom parts among suppliers in make to order manufacturing is formulated as a single- or multi-objective mixed integer program. Given a set of customer orders for products, the decision maker needs to decide from which supplier to purchase custom parts required for each customer order. The selection of suppliers is based on price and quality of purchased parts and reliability of on time delivery. The risk of defective or unreliable supplies is controlled by the maximum number of delivery patterns (combinations of suppliers delivery dates) for which the average defect rate or late delivery rate can be unacceptable. Furthermore, the quantity or business volume discounts offered by the suppliers are considered. Numerical examples are presented and some computational results are reported.

مقدمه انگلیسی

In make-to-order environment customer-oriented manufacturers should be prepared to produce varieties of products to meet the different customer needs. Each product is typically composed of many common and non-common (custom) parts that can be sourced from different approved suppliers with different supply capacity. An important issue is how to best allocate the orders for parts among various part suppliers to fulfill all customer orders for products and to achieve a high customer service level at a low cost. The decision maker needs to decide from which supplier to purchase parts required to complete each customer order. The above decisions are based on price, quality (defect rate) and reliability (on time delivery) criteria that may conflict each other, e.g. the supplier offering the lowest price may not have the best quality or the supplier with the best quality may not deliver on time. Furthermore, to reduce the fixed ordering (transaction) costs the number of suppliers and the total number of orders should be minimized. On the other hand, the selection of more suppliers sometimes may divert the risk of unreliable supplies.
In spite of the importance of supplier selection and order allocation problems, the decision making is not sufficiently addressed in the literature (for a recent review, see Aissaoui et al. [1]), in particular for make-to-order manufacturing environment, e.g. Murthy et al. [2], Sawik [3], Yue et al. [4]. Basically, the authors distinguish between single and multiple item models and supplier selection with single or multiple sourcing, where each supplier can fully meet all requirements (e.g. Akinc [5]) or none of the suppliers is able to satisfy the total requirements, respectively. The vast majority of the decision models are mathematical programming models either single objective, e.g. Kasilingam and Lee [6], Basnet and Leung [7], Jayaraman [8] or multiple objectives, e.g. Weber and Current [9], Xia and Wu [10], Demirtas and Ustun [11], Ustun and Demirtas [12], Pokharel [13].
The supplier selection is a complex decision making problem which includes both quantitative and qualitative factors and one of the disadvantages of the mathematical programming methods is their failure to account for qualitative factors that may affect suppliers performance. In order to consider both quantitative and qualitative factors some researchers propose hybrid approaches that combine different methods. For example, Sanayei et al. [14] propose an integration of multi-attribute utility theory and linear programming, first to rate and choose the best suppliers and then to find optimal allocation of order quantities among the selected suppliers to maximize total additive utility. The combined method allows both quantitative and qualitative factors under risk and uncertainty to be considered as well as to account for the probabilistic nature of supplier performance. Another integrated approach that combines analytic network process and multi-objective mixed integer programming is proposed in [11] and [12]. First, the potential suppliers are evaluated according to 14 criteria that are involved in the four clusters: benefits, opportunities, costs and risks, to calculate the priorities of each supplier. Then, the optimum quantities are allocated among selected suppliers to maximize total value of purchasing (using the calculated priorities) and to minimize the total cost and total defect rate. However, the disadvantages of the integrated methods usually may affect the performance of hybrid approaches. The other approaches that are also applied to solve the supplier selection problem are methods based on fuzzy sets, e.g. a fuzzy multi-objective integer programming Huo and Wei [15] and genetic algorithms (e.g. Liaoa and Rittscher [16], Che and Wang [17]). For example, in [16] a genetic algorithm with problem specific operator is developed to account for the inbound transportation and to combine supplier selection with carrier selection decisions. The fuzzy and genetic algorithms, however, are heuristics that do not guarantee optimality of a solution.
The models developed for supplier selection and order allocation can be either single-period models (e.g. [6], [8], [9] and [11]) that do not consider inventory management or multi-period models (e.g. [3], [7], [12] and [16], Ghodsypour and O’Brien [18], Tempelmeier [19]) which consider the inventory management by lot-sizing and scheduling of orders. Since common parts can be efficiently managed by material requirement planning methods, this research is focused on custom parts that can be critical in make-to-order manufacturing. For custom-engineered products no inventory of custom parts can be kept on hand. Instead, the custom parts need to be requisitioned with each customer order and hence the custom parts inventory need not to be considered.
This paper presents mixed integer programming models for single or multiple objective supplier selection in make-to-order manufacturing for a static supply portfolio in a non-discount or discount environment, that is for the allocation of orders for parts among the suppliers without or with discount and with no timing decisions. In contrast to the dynamic portfolio, which is the allocation of orders among the suppliers combined with the allocation of orders among the planning periods.
The major contribution of this paper is that it proposes a simple mixed integer programming approach for selection of supply portfolio under conditions of operational risk associated with uncertain quality and reliability of supplies. The integer programming models incorporate risk constraints where the risk of defective or unreliable supplies is controlled by the maximum number of delivery patterns (combinations of suppliers delivery dates) for which the average defect rate or late delivery rate may exceed the maximum acceptable rates. The number of maximum delivery patterns and the corresponding maximum rates represent, respectively, the confidence level and the targeted rates above which a risk averse decision maker wants to limit the number of outcomes.
The paper is organized as follows. In Section 2 description of the supplier selection problem in make-to-order manufacturing is provided. The mixed integer program for a single objective supplier selection in a non-discount environment is presented in Section 3. The model enhancements for the supplier selection with a business volume discount or quantity discount are presented in Section 4. The multiple objective approach is proposed in Section 5. Numerical examples and some computational results are provided in Section 6, and final conclusions are made in the last section.

نتیجه گیری انگلیسی

The problem of optimal allocation of orders for parts among a set of approved suppliers in make-to-order manufacturing has been modeled as a mixed integer program, in which two different objective functions were either aggregated into a single cost function to be minimized or considered as independent functions of a dual objective optimization problem. In particular, the risk level of defective or unreliable supplies that the decision maker is disposed to accept is controlled by the maximum number View the MathML sourcev¯ of delivery patterns for which the average defect rate or late delivery rate can be unacceptable. Under the assumption of identical probability associated with each of the h delivery patterns, the decision parameters View the MathML sourceq¯ and View the MathML sourcer¯ can be considered as the targeted average defect and late delivery rates based on the View the MathML source(1-v¯/h)-percentile of these rates. In other words, we allow View the MathML source100(v¯/h)% of the delivery patterns to exceed View the MathML sourceq¯ and/or View the MathML sourcer¯, and the smaller is View the MathML sourcev¯, the more risk averse is the selected supply portfolio.
In a single objective approach the corresponding costs of defective or delayed parts are directly introduced in a minimized cost function, while in a multiple objective approach the average defect rate and the average late delivery rate are aggregated and considered as a separate objective function to be minimized.
A single objective approach is capable of finding a single optimal portfolio of suppliers that minimizes the total average cost of ordering, purchasing, defects and delays of parts. The proposed aggregation of the different types of cost into a single cost function should account for a specific business environment and the decision maker preferences. In practice, cost of a defective or delayed part may sometimes exceed its price. In particular, when the resulting lack of all required parts leads to a much higher cost of unfulfilled customer orders. On the other hand, a subset of nondominated portfolios can be found, applying the multiple objective Tchebycheff approach to minimize the average ordering and purchasing cost of parts and the average defect and late delivery rate of supplies.
The computational results indicate that while the single objective approach yields a single proven optimal solution in a very short CPU time, the multiple objective approach requires the much longer computation time to produce a very small subset of nondominated solutions. In addition, a single objective solution weakly dominates some of the nondominated multiple objective solutions, which further justifies the usefulness of the proposed single objective approach.
In the models proposed various simplifying assumptions have been introduced. For example, it has been assumed that each supplier is capable of manufacturing all required part types. In a more general setting, each supplier may only be prepared to manufacture a subset of part types and provide with the parts the corresponding subset of customer orders. Furthermore, the quantity discount offered by the supplier has been assumed to be based on the total quantity of all ordered parts. In practice, independent quantity discounts may be offered for the individual part types.
The limited computational experiments have indicated that the proposed approach requires a relatively small CPU time to find the optimal solution in a static case, where all customer orders are known ahead of time. The last assumption can be relaxed, and the approach can also be used in a dynamic case where orders arrive irregularly over time. In this case, the supplier selection decisions can be made periodically over a rolling planning horizon, upon arrivals of a number of orders in a specific time interval.